A Heuristic Algorithm to Optimise Stope Boundaries

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2000

A heuristic algorithm to optimise stope boundariesMajid Ataee-PourUniversity of Wollongong

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Recommended CitationAtaee-Pour, Majid, A heuristic algorithm to optimise stope boundaries, Doctor of Philosophy thesis, Faculty of Engineering,University of Wollongong, 2000. http://ro.uow.edu.au/theses/2923

http://ro.uow.edu.au/http://ro.uow.edu.au/http://ro.uow.edu.auhttp://ro.uow.edu.au/theseshttp://ro.uow.edu.au/thesesuowhttp://ro.uow.edu.au/http://ro.uow.edu.au/

A HEURISTIC ALGORITHM TO OPTIMISE

STOPE BOUNDARIES

A thesis submitted in fulfilment of the requirements

for the award of the degree

DOCTOR OF PHILOSOPHY

from

UNIVERSITY OF WOLLONGONG

by

MAJID ATAEE-POUR

B.Eng., M.Eng. (Hons.) Mining Engineering

Faculty of Engineering

March 2000

IN THE NAME OF GOD

This thesis is dedicated to

the soul of

m y dear mother;

Sedigheh Alipour-Hejranian

for her love during her life.

n

DECLARATION

This thesis has been purposefully and originally undertaken by the author for a degree of

Doctor of Philosophy in the department of Civil, Mining and Environmental

Engineering at the University of Wollongong. This thesis contains no material

submitted, in whole or in part, for a degree at this or any other institution. The following

publications have been based on this thesis:

Ataee-pour, M, 1997, A New Heuristic Algorithm to Optimise Stope Boundaries,

Proceedings of the Second Regional A P C O M Symposium on Computer Applications

and Operations Research in the Mineral Industry, L A Puchkov (ed.), Moscow, Russia,

6 p.

Ataee-pour, M and Baafi E Y, 1998, Implementation of a Heuristic Algorithm to

Optimise Stope Limits With Excel Modules, Proceedings of the Third Regional A P C O M

Symposium on Computer Applications in the Mineral Industries, Basu et al (eds.), The

Australasian Institute of Mining and Metallurgy, Kalgoorlie, Autralia, pp. 161-164.

Ataee-pour, M and Baafi E Y, 1999, Stope Optimisation Using the Maximum Value th

Neighbourhood (MVN) Concept, Proceedings of the 28 International Symposium on

Computers Applications in the Minerals Industries, K Dagdelen (ed.), Colorado, pp.

493-501.

Ataee-pour, M and Baafi E Y, 2000, Visual Basic Implementation of the Maximum

Value Neighbourhood Algorithm to Optimise Stope Boundaries. Paper accepted for

presentation at the Ninth International Symposium on Mine Planning and Equipment

Selection, "MPES'2000", Greece.

MAJID ATAEEPOUR

in

A CKNO WLEDGEMENT

I a m very grateful for the helpful contributions made by a number of people during the

course of this study. I wish to express m y sincere thanks to:

Associate Professor E Y Baafi, Faculty of Engineering, University of Wollongong, for

his supervision of the thesis and his advice, encouragement, guidance and critical

review during the entire course of this study;

Dr. P Standish, Elura mine, Cobar, NSW, for allowing the visit to the Elura mine at

Cobar;

Mrs. K Draisma, Learning Development Centre, University of Wollongong, for her

guidance in thesis writing and Ms. J Shaw for her proofreading of the thesis; and

All staff members in the Faculty of Engineering, University of Wollongong, for

providing computer facilities, support and help during this work.

I wish to, gratefully, acknowledge the Ministry of Culture and Higher Education of the

Islamic Republic of Iran, for the financial support and sponsorship through the period of

this project.

Last, but not the least, I would like to express my sincere appreciation to my wife Zahra,

m y daughter Maedeh and all members of m y family in Iran, for their patience and

acceptance of hardship during the entire course of m y study and for their understanding,

which has been much more than I could explain.

IV

ABSTRACT

Determination of the optimum mine layout is one of the important tasks in mine

planning. In the case of open pit mines, a large number of algorithms using a range of

techniques have been developed to generate the true optimum solution. Several

commercial computer packages are available to assist mining engineers design open

pits. In contrast, only a few algorithms, using limited techniques, have been developed

to optimise the stope geometry in underground operations. Most of which fail to provide

optimum 3D solutions.

A heuristic algorithm, termed the "Maximum Value Neighbourhood" (MVN) was

developed in this thesis to optimise stope boundaries. The MVN algorithm benefits from

its simplicity in both concept and implementation. It provides a 3D analysis and can be

applied to any underground mining method, although it does not guarantee the true

"optimum" stope layout. The MVN algorithm uses a 3D fixed economic block model to

locate the best neighbourhood of a block, which guarantees the maximum net value.

Neighbourhoods are restricted by the mine geometry constraints. The neighbourhood

concept is based on the number of mining blocks equivalent to the minimum stope size.

Since a variety of neighbourhoods are available for each block, the one that provides the

maximum net value (the maximum value neighbourhood, MVN) is located for inclusion

in the final stope.

In order to test the algorithm, the 3D version of the MVN algorithm was implemented on

small sized examples, using the Visual Basic for Applications (VBA) modules supported

by Microsoft Excel. The framework of the Excel worksheets was suitable to store block

data and display the optimised stope.

A Fortran 90 program, the "Stope Limit Optimiser" (SLO), was developed to

implement the 3 D MVN algorithm on actual mine data. The SLO optimiser integrates

v

the Fortran 90 code of the algorithm with the Winteracter user interface features, to

provide dialog boxes and user friendly menus. The SLO provides a Windows based

interactive environment to define and edit the project parameters including the block

model parameters, the stope geometry constraints and the economic factors.

VI

TABLE OF CONTENTS

DECLARATION iii

ACKNOWLEDGEMENTS iv

ABSTRACT v

TABLE OF CONTENTS vii

LIST OF FIGURES xiii

LIST OF TABLES xix

LIST OF SYMBOLS AND ABBREVIATIONS xx

CHAPTER ONE: INTRODUCTION

1.1 GENERAL INTRODUCTION 1-1

1.2 STATEMENT OF THE PROBLEM 1-3

1.2.1 Status of Pit Limit Optimisation 1-4

1.2.2 Status of Stope Geometry Optimisation 1-6

1.3 SCOPE OF THE THESIS 1-9

1.4 OUTLINE OF THE THESIS 1-11

CHAPTER TWO: ULTIMATE MINE DESIGN METHODS

2.1 INTRODUCTION 2-1

2.2 MINE G E O M E T R Y OPTIMISATION 2-1

2.2.1 Optimisation Criteria 2-2

2.2.2 Problem Formulation 2-3

2.2.3 Necessity of Optimisation Algorithms 2-4

2.3 ULTIMATE PIT OPTIMISATION ALGORITHMS 2-5

2.3.1 Moving Cone Algorithms 2-7

2.3.2 Dynamic Programming Algorithms 2-11

2.3.3 Graph Theory Algorithms 2-16

vii

2.3.4 Network Flow Analysis 2-19

2.3.5 Other Approaches 2-21

2.4 STOPE LIMIT OPTIMISATION ALGORITHMS 2-24

2.4.1 Dynamic Programming Algorithm 2-25

2.4.2 Geostatistical Approach 2-29

2.4.3 Octree Division Algorithm 2-31

2.4.4 Floating Stope Algorithm 2-38

2.4.5 Branch and Bound Technique 2-41

2.5 ORE-BODY BLOCK MODEL 2-48

2.5.1 Types of Block Model 2-50

2.5.2 Block Size 2-52

2.5.3 Estimation Process 2-54

2.6 ECONOMIC BLOCK MODEL 2-56

2.6.1 Rules of Thumb in Calculating Block Values 2-58

2.6.2 Block Valuation 2-59

2.7 S U M M A R Y 2-60

CHAPTER THREE: MAXIMUM VALUE NEIGHBOURHOOD (MVN)

ALGORITHM


3.2 BASIC SPECIFICATIONS OF STOPE GEOMETRY 3-1

3.3 FORMULATION OF THE PROBLEM 3-4

3.4 STOPE GEOMETRY CONSTRAINTS 3-6

3.5 THE NEIGHBOURHOOD CONCEPT 3-8

3.5.1 The Neighbourhood Set 3-10

3.5.2 The Optimum Neighbourhood 3-14

3.5.3 Infeasible Neighbourhoods 3-18

3.6 THE MVN ALGORITHM 3-18

3.6.1 The Optimisation Procedure 3-19

3.6.2 Numerical Examples 3-25

viii

3.6.3 Algorithm Capability to Recover the Stope 3-34

3.7 SUMMARY 3.36

CHAPTER FOUR: THE 2D AND 3D MAXIMUM VALUE NEIGHBOURHOOD

ALGORITHMS

4.1 INTRODUCTION 4_j

4.2 TWO- AND THREE-DIMENSIONAL NEIGHBOURHOODS 4-1

4.2.1 The Order of Neighbourhood 4-6

4.2.2 The Optimum Neighbourhood 4-8

4.2.3 Infeasible Neighbourhoods 4-12

4.3 A 2D NUMERICAL EXAMPLE 4-15

4.4 IMPLEMENTATION OF 3D MVN ALGORITHM USING VBA CODE 4-23

4.5 SUMMARY 4-40

CHAPTER FIVE: IMPLEMENTATION OF THE MAXIMUM VALUE

NEIGHBOURHOOD ALGORITHM


5.2 THE SLO PROGRAM COMPONENTS 5-1

5.3 THE WINTERACTER RESOURCE SCRIPT DESCRIPTION 5-2

5.3.1 The SLO Menus 5-3

5.3.2 The SLO Dialogs 5-4

5.4 THE FORTRAN 90 SOURCE CODE DESCRIPTION 5-5

5.4.1 The SLO Input/Output Files 5-5

5.4.2 The SLO Main Program 5-6

5.4.3 The SLO Sub-programs 5-8

5.5 THE SLO GENERAL PROCEDURE 5-9

5.6 THE SLO PROJECTS 5-12

5.7 PROJECT DEFINITION FILES 5-14

5.7.1 Block Model Parameters 5-15

ix

5.7.2 Stope Geometry Constraints 5-21

5.7.3 Economic Factors 5_25

5.8 BLOCK DATA FILE 5.34

5.9 DATA PREPARATION 5.37

5.9.1 Equivalent Grade 5.38

5.10 S U M M A R Y 5.39

CHAPTER SIX: PROGRAMMING OF THE MAXIMUM VALUE

NEIGHBOURHOOD ALGORITHM IN FORTRAN 90


6.2 GENERAL VIEW ON THE OPTIMISATION STAGE 6-1

6.3 DEFINING THE OPTIMISATION DATA 6-5

6.4 LOCATING THE M A X I M U M VALUE NEIGHBOURHOOD 6-8

6.4.1 Neighbourhood Identification 6-10

6.4.2 Neighbourhood Determination 6-14

6.4.3 Coding the Procedure 6-19

6.4.4 Feasibility and Valuation of Neighbourhoods 6-20

6.5 UPDATING THE STOPE 6-23

6.6 S U M M A R Y 6-26

CHAPTER SEVEN: OUTPUT RESULTS OF THE STOPE LIMIT OPTIMISER


7.2 BLOCK FLAG DATA 7-2

7.3 DISPLAY OF RESULTS 7-2

7.3.1 Designing the Borders of the Plan /Section 7-7

7.3.2 Designing the Body of the Plan / Section 7-8

7.4 REPORTS 7-9

7.4.1 Intermediate Results 7-9

7.4.2 Neighbourhood Results 7-13

X

7.4.3 Summary Report 7-14

CHAPTER EIGHT: VALIDATION OF THE STOPE LIMIT OPTIMISER


8.2 OPTIMISER VALIDATION 8-1

8.3 THE VALIDATION PROCEDURE 8-2

8.3.1 Manual Validation 8-2

8.3.2 An Example with a Special Pattern 8-4

8.3.3 Validation with Excel VBA Modules 8-6

8.3.4 Program Debugging 8-6

CHAPTER NINE: SUMMARY AND CONCLUSIONS

9.1 SUMMARY 9-1

9.2 CONCLUSIONS 9-4

9.3 RECOMMENDATION AND FUTURE WORKS 9-6

REFERENCES

APPENDIX A: COMPUTATIONS OF BLOCK ECONOMIC VALUES

A.l Block Valuation A-l

A.2 Equivalent Grade A-5

APPENDIX B: SLO DIALOGS AND SUBPROGRAMS

APPENDIX C: THE STOPE LIMIT OPTIMISER USER'S GUIDE

C.l INTRODUCTION C-l

C.2 INSTALLATION C-l

C.3 STARTING SLO C-2

XI

C.4 PROJECT MANIPULATION C-3

C.4.1 Project \ New option C-4

C.4.2 Project \ Open option C-5

C.4.3 Project \ Close option C-5

C.4.4 Project \ Save option C-6

C.4.5 Project | Save as... option C-6

C.4.6 Project \ Exit option C-7

C.5 INPUT EDITION C-7

C.5.1 Edit | Block Model option C-8

C.5.2 Edit | Sub-regions option C-l 1

C.5.3 Edit | Stope Constraints option C-13

C.5.4 Edit | Economic Factors option C-15

C.5.5 Edit | Main Data option C-20

C.6 PRE OPTIMISATION C-21

C.6.1 Preoptimisation | Data Preparation Option C-21

C.6.2 Preoptimisation | Select Region Option C-22

C.6.3 Preoptimisation | Import Block Data Option C-23

C.7 OPTIMISATION C-23

C.7.1 Run | Optimise Option C-23

C.8 POST OPTIMISATION C-24

C.8.1 Results | Export Flag Data Option C-24

C.8.2 Results | Summary Report Option C-25

C.8.3 Results | Intermediate Results Option C-25

C.8.4 Results | Neighbourhood Results Option C-25

C.8.5 Results | Test Results Option C-26

C.8.6 Results | Plot Plans/Sections Option C-26

C.8.7 Results | View Plots Option C-27

C.9 HELP C-28

C.9.1 Help | About SLO Option C-28

Xll

LIST OF FIGURES

Figure Page

1.1 The Mining Industry Share of the Australian Export Dollar 1986-1998 1-2

(Source: Australian Bureau of Statistics, 1998)

1.2 The Significance of Metallic Minerals in the Australian Mineral 1 -2 Products (Source: Australian Bureau of Statistics, 1998)

1.3 A Schematic View of the Proposed Study 1-10

2.1 Variation of the Total Value Compared to the Unit Value of a 2-3 Specific Mine over the Mine Production (Wilke et al, 1984)

2.2 The Moving Cone Technique 2-9

2.3 Failure of the Moving Cone to Recognise the Maximum Value Pit 2-10

2.4 Failure of the Moving Cone Algorithm to Recognise a Positive Value 2-10

Pit (Wright, 1990)

2.5 A Dynamic Programming Technique Applied to Pit Limit 2-13

Optimisation

2.6 Graph Theory Applied to Pit Limit Ooptimisation 2-18

2.7 The Network Flow Technique Applied to Pit Limit Optimisation 2-20

2.8 Comparison between Open Pit and Block-Caving (Riddle, 1977) 2-26

2.9 Adjacent Blocks in Open Pit and Block-Caving 2-27

2.10 Conditional Simulation of Lenses (Deraisme and de Fouquet, 1984) 2-30

2.11 Geometrical Constraints for the Three Mining Methods at Ben 2-30

Lomond Mine (Deraisme et al, 1984)

2.12 Compared Outlines for the Cut-and-Fill Method on a Section 2-31

(Deraisme era/., 1984)

2.13 Successive Removal of Sub-Volumes 2-33

2.14 A n Example of Quadtree Division 2-34

2.15 Octree Division 2-34

2.16 A n Example of Vein Surrounded by an Initial Mineable Volume, 2-35

Resulting from the Object Manipulator (Cheimanoff et al, 1989)

2.17 Successive Sub-Volume Evaluation Steps Performed by Shape 2-37 Generator, on a Cut-and-Fill Method (Cheimanoff et al, 1989)

2.18 Joint Consideration of Sub-Volumes 2-37

2.19 The "Inner" and the "Outer" Envelopes for a Single Block 2-39

Xlll

LIST OF FIGURES

Figure Page

1.1 The Mining Industry Share of the Australian Export Dollar 1986-1998 1-2 (Source: Australian Bureau of Statistics, 1998)

1.2 The Significance of Metallic Minerals in the Australian Mineral 1 -2 Products (Source: Australian Bureau of Statistics, 1998)

1.3 A Schematic View of the Proposed Study 1-10

2.1 Variation of the Total Value Compared to the Unit Value of a 2-3 Specific Mine over the Mine Production (Wilke et al, 1984)

2.2 The Moving Cone Technique 2-9

2.3 Failure of the Moving Cone to Recognise the Maximum Value Pit 2-10

2.4 Failure of the Moving Cone Algorithm to Recognise a Positive Value 2-10 Pit (Wright, 1990)

2.5 A Dynamic Programming Technique Applied to Pit Limit 2-13 Optimisation

2.6 Graph Theory Applied to Pit Limit Ooptimisation 2-18

2.7 The Network Flow Technique Applied to Pit Limit Optimisation 2-20

2.8 Comparison between Open Pit and Block-Caving (Riddle, 1977) 2-26

2.9 Adjacent Blocks in Open Pit and Block-Caving 2-27

2.10 Conditional Simulation of Lenses (Deraisme and de Fouquet, 1984) 2-30

2.11 Geometrical Constraints for the Three Mining Methods at Ben 2-30 Lomond Mine (Deraisme et al., 1984)

2.12 Compared Outlines for the Cut-and-Fill Method on a Section 2-31

(Deraisme et al, 1984)

2.13 Successive Removal of Sub-Volumes 2-33

2.14 A n Example of Quadtree Division 2-34

2.15 Octree Division 2-34

2.16 A n Example of Vein Surrounded by an Initial Mineable Volume, 2-35

Resulting from the Object Manipulator (Cheimanoff et al, 1989)

2.17 Successive Sub-Volume Evaluation Steps Performed by Shape 2-37 Generator, on a Cut-and-Fill Method (Cheimanoff et al, 1989)

2.18 Joint Consideration of Sub-Volumes 2-37

2.19 The "Inner" and the "Outer" Envelopes for a Single Block 2-39

Xlll

2.20 Limitations of the Moving Cone and Floating Stope Methods 2-41

2.21 Typical Piecewise Linear Approximation Function 2-43

2.22 Cumulative Block Value Function Applied to a R o w of Blocks 2-44

2.23 SOS2 Solution Example 2-47

2.24 The Ore-Body Surrounded by a large box 2-49

2.25 The Ore-Body Surrounded by the Physical Block Model 2-49

2.26 A n Example of a Geological Unit Coded into a Block Model (Noble, 2-50 1992)

2.27 Modified Block Models Compared to a Fixed Block Model 2-51 (Badiozamani and Roghani, 1988)

3.1 Geometric Constraints, Open Pit versus Underground Mines 3-2

3.2 Four Examples of Possible Choices (stopes) for Mining the Bblock, 3-3

By

3.3 Block Dependence, Open Pit versus Underground Mining 3-4

3.4 The Spatial Address of an Economic Block 3-5

3.5 Stope Constraints: ( a) ID, ( b ) 2D and ( c ) 3D Problems 3-7

3.6 Minimum Stope Size versus the Neighbourhood (NB) 3-9

3.7 Possible Neighbourhoods of the Block, B4, for NB Orders of 2, 3 and 3-11 4

3.8 A Typical R o w of Blocks with "n" Blocks 3-11

3.9 Locating the Maximum Value Neighbourhood 3-16

3.10 Infeasible Neighbourhoods 3-19

3.11 Generalised Flow-Chart for the Optimisation Procedure 3-22

3.12 A Simplified Pseudo Code for the MVN Algorithm 3 -24

3.13 A R o w of Blocks with 10 Columns taken from an Economic Block 3-25

Model

3.14 The Optimised Example Manually using the MVN Algorithm 3-31

3.15 Optimising a Stope Section using the MVN Algorithm with a ID 3-32

(height) Constraint

4.1 Examples of 2D Possible Neighbourhoods for a Specific Block, B33 4-3

4.2 A Typical Block Model illustrating 3D Stope Size Constraints 4-5

4.3 Examples of 3D Neighbourhoods 4-5

4.4 Possible 2D Neighbourhoods of the Block, B44, with [Onb](2) = (3,3) 4-7

4.5 Possible 3D Neighbourhoods of the Block, % , with [Onb](3) = (2, 2, 4-8

2)

XIV

.6 Overlaid 3D Neighbourhoods for the Block, Bijk, with the Order of (2, 4-9 2,2)

4.7 The mth Neighbourhood of the Block, B0, with any 2D Order of 4-10 Neighbourhood

4.8 Feasibility of 2D Neighbourhoods for Boundary Blocks where 4-13 [Onb](2) = (2, 2)

4.9 Feasibility of 3D Neighbourhoods for Boundary Blocks where 4-14 [Onb](3) = (2, 2, 2)

4.10 A Section of an Economic Block Model with 6 Rows and 10 4-15 Columns

4.11 Results of Applying the 2D MVN Algorithm to the First Two 4-22 Columns of the Given Example Section

4.12 The Section Optimised with the 2D MVN Algorithm 4-22

4.13 Flow- Chart of the Main Program Written in VBA 4-25

4.14 The Main Menu of the VBA Program to Implement the MVN 4-26 Algorithm

4.15 The Dialog Box to Input Specifications of the Model 4-27

4.16 A Small 3D Economic Model Example 4-28

4.17 The Dialog Box to the Input Stope Geometry Constraints 4-29

4.18 The Dialog Box Providing Options for Reading Economic Block Data 4-30

4.19 Flow-Chart of the Subroutine "MaxNbor" 4-31

4.20 The "Perform Optimisation" Window 4-35

4.21 The Optimised Stope Geometry of Section 2 4-37 (Sheet 2, in the Excel environment)

4.22a Optimum Stope Example, [k = 2, Onb = (3, 3, 3)] 4-39

4.22b Optimum Stope Example, [k = 2, Onb = (2, 3, 4)] 4-39

4.22c Optimum Stope Example, [k = 2, Onb = (3, 2, 3)] 4-39

5.1 Example Source Code of the Main Program of the SLO Program 5-9

5.2 Generalised Flow-Chart of the Stope Limit Optimiser (SLO) 5-11

5.3 The "Project" Option from the Main Menu in the SLO Program 5-13

5.4 The SLO Dialog for Opening an Existing Project 5-14

5.5 Mode Selection for Block Model Input 5-15

5.6 The " X Y Z Mode" Dialog Box for Defining the Block Model 5-16

5.7 The Dialog Box for Defining Sub-Regions in the Block Model 5-17

5.8 Sub-Regions Dialog Box 5-18

5.9 Defining Sub-Regions within the Block Model 5-19

XV

5.10 Example of the Model Parameters File," .mpr"

5.11 The Dialog Box for Defining the Order of Neighbourhood (entire model)

5.12 The Dialog Box to Select a Sub-Region for Defining it's Order of Neighbourhood

5.13 The Dialog Box for Defining the Order of Neighbourhood (sub-regions)

5.14 Example of the Stope Geometry Constraints File," .est"

5.15 The SLO Dialog for Defining Products of the Project

5.16 The SLO Dialog for Defining Prices and Price Units

5.17 The Dialog Box for Defining Various Costs

5.18 The Dialog Box for Defining Various Rates of Recovery for the Main

Product

5.19 The Dialog Box for Defining Various Rates of Recovery of By-products

5.20 The SLO Dialog Box for Defining Grade Units

5.21 The Dialog Box for Defining Ore Properties

5.22 Example of the Economic Factors File, ".eco"

5.23 The Dialog Box for Defining the (Main) Data File

5.24a Example of the Assay Value Data File, ".dat"

5.24b Example of the Economic Value Data File," .dat"

5.25 Preparation of the Main Data File for the Optimisation Core

6.1 Major Modules Used to Perform the Optimisation Algorithm

6.2 The SLO Dialog Box which Specifies the Domain of the Optimisation

6.3 Numbering the Elements of a 2D Neighbourhood, [Onb](2) = (4, 3)

6.4 Numbering the Elements of a General 2D Neighbourhood (numbers inside the cells indicate the sequential ID number of that element

within the NB.)

6.5 Numbering the Elements of a 3D Neighbourhood, [Onb](3) = (4, 2, 3)

6.6 The Identification of 2D Neighbourhoods

6.7 Two Neighbourhoods for the Block, Bijk, with the Order of (2, 2, 2)

6.8 A 2D Infeasible Neighbourhood (it is known that the element, BEVap,

is outside the model.)

6.9 The General Flow-Chart of the "SelectBlock" Subroutine

7.1 Example of the Output File, ".out" (the block model is 4 x 3 x 3)

7.2 SLO Dialog Box for Plotting Specified Plans or Sections

XVI

7.3 The Orientation of Plots in the SLO Program

7.4 Example of the Plans/Sections Plotted in the SLO Program

7.5 Main Subroutines Called to Plot any Plan or Section

7.6 Example of the Intermediate Optimisation Results Collected in the "*.res" Files

7.7 Example of the Neighbourhood Results Collected in the "*.nbv" Files

7.8 A Screen Summary Report

8.1 A Simple Block Model

8.2 Optimised Stope Example (using SLO)

8.3 Optimised Stope Example (manual technique)

8.4 Dollar Value Data of the Block Model Repeated in all Levels

8.5 The Optimised Stope Plot in all Levels (plans)

8.6 The Optimised Stope Section Obtained in the Excel Worksheets

8.7 Example of the Information Stored in "*.tst" Files Related to the Block Assay Information

8.8 Example of the Block Economic Data Stored in a "*.tst" File

8.9 Example of the Optimisation Process Results stored in "*.tst" Files

8.10 Example of the Summary Results Stored in "*.tst" Files

A. 1 Block value function

C.l The Winteracter Based Welcome Page of the Stope Limit Optimiser (SLO)

C.2 Options Available for the Project Menu Item

C.3 The SLO Dialog to Specify N e w Projects

C.4 The SLO Dialog for Opening an Existing Project

C.5 The SLO Dialog for Ssaving the Current Project with a N e w Name/Directory

C.6 Options Available from the Edit Menu Item

C.7 Mode Selection for Block Model Input

C.8 The SLO Dialog which allows the Definition of the Block Model in

the X Y Z Mode

C.9 The SLO Dialog which allows the Definition of the Block Model in

the U K Mode

C I O The Dialog Box for Choosing Options in the Block Model

C. 11 The SLO di Dialog for Manipulating Sub-Regions

C. 12 Defining/ Editing a Sub-Region within the Block Model

XVll

C. 13 The SLO Dialog for Defining the Stope Geometry Constraints (entire C-14 model)

C. 14 The SLO Dialog for Defining the Stope Geometry Constraints (sub- C-14 regions)

C.15

C.16

C.17

C.18

C.19

C.20

C.21

C.22

C.23

C.24

C.25

C.26

C.27

The Dialog for Defining the Products of the SLO Project

The Dialog for Defining Prices and Price Units in the SLO Program

The Dialog for Defining Costs in the SLO Program

The SLO Dialog for Defining the Rates of Recovery (main product)

The SLO Dialog for Defining the Rates of Recovery (by-products)

The SLO Dialog for Defining Grade Units

The SLO Dialog for Defining Ore Properties

The SLO Dialog for Defining the (main) Data File

Options Available for the Preoptimisation Menu Item

The SLO Dialog for Defining the Optimisation Domain

Options Available for the Results Menu Item

Example of the SLO Summary Report

The SLO Dialog for Plotting the Optimum Stope

C-16

C-16

C-17

C-18

C-19

C-19

C-20

C-21

C-22

C-23

C-24

C-25

C-26

XVlll

LIST OF TABLES

Table

1.1

1.2

1.3

3.1

3.2

4.1

5.1

9.1

A.l

A.2

A.3

B.l

B.2

B.3

Value of Australian Minerals Produced (1992-1997)

Papers and their Contributions to Mine Geometry Optimisation (open pit vs. underground)

Reports on Mine Geometry Optimisation (by subject)

Possible Neighbourhoods of the Block, Bj, for different "Onb"

Summary of the Algorithm Applied for Column 3

Results Summary for Blocks in Section 2 of the Example

Description of Input/Output Files used in the SLO Program

Capabilities and Restrictions of Stope Geometry Optimisation Algorithms

Equivalent Grades Calculated for a Deposit with Two By-Products A-7

Grade Factors Applied for Corrections in MPEG Formulae A-8

Price Factors Applied for Corrections in MPEG Formulae A-8

Dialogs Used in the SLO Program B-2

List of Subroutines Defined in the SLO Source Code B-5

List of Functions Defined in the SLO Source Code B-12

XIX

LIST OF SYMBOLS AND ABBREVIA TIONS

$b: Billion dollars

$m: Million dollars

[NB](2): Two-dimensional neighbourhood

[NB](3): Three-dimensional neighbourhood

[NBS](2): Set of 2D neighbourhoods

[NBS](3): Set of 3D neighbourhoods

[Onb] ' Two-dimensional order of neighbourhood

[Onb](3>: Three-dimensional order of neighbourhood

ID: One-dimensional

2D: Two-dimensional

3D: Three-dimensional

Al: Artificial Intelligence

ANN: Artificial Neural Network

ASCII: American Standard for Communication Interchange

BEV: Block Economic Value

Byk'. the block located at the ith ww,j'h column and k*h section

BMC: Block Mining Cost

BRR: Block Revenue Ratio

C A D : Computer Aided Drafting

CM'- unit cost of metal extraction from the ore

Core- unit cost of ore (waste) extraction from the deposit

DP: Dynamic Programming

EF: Equivalence Factor

F: Flag of blocks

g: grade

GA: Genetic Algorithms

gc: cut-off grade

GT: Graph Theory

XX

GV: Gross Value

I: the number of blocks in the X direction of the block model

i: the block number in the X direction of the block model

J: the number of blocks in the Y direction of the block model

j: the block number in the Y direction of the block model

K: the number of blocks in the Z direction of the block model

k: the block number in the Z direction of the block model

kg: kilogram

LG: Lerchs-Grossmann

LP: Linear Programming

m: metre

m2: square metre

m3: cubic metre

Max: maximum

Min: minimum

MNB V: Maximum Neighbourhood Value

MPEG: Main Product Equivalence Grade

MVN: Maximum Value Neighbourhood

n(A): the size (number of elements) of a given set, A

NA: Not Applicable

NB: Neighbourhood

NBS: Set of Neighbourhoods

NBV: Neighbourhood Value

NB VS: Set of Neighbourhood Values

N S W : N e w South Wales

Onb ' order of neighbourhood

oz: ounce

P: price

ppm: part per million

r: recovery

SBR: Stope Block Ratio

SEV: Stope Economic Value

XXI

Stope Limit Optimiser

Type-Two Special Ordered Sets

tonne

universal set

volume

Visual Basic for Applications

density

xxn

CHAPTER ONE

INTRODUCTION

1.1 GENERAL INTRODUCTION

The mining industry is a significant contributor to Australia's export dollars. Figure 1.1

illustrates the mining industries contribution to Australia's total export dollars from

1986 to 1998. The current trend shows that mineral product exports have grown more

slowly than that total exports. However, the mining sector is still supplying about one

quarter of the country's total exports.

Mineral products consist of three groups, which are metallic minerals, coal and oil and

gas. During the year ended June 1997, the total value of minerals produced in Australia

was $Aus 31.4 billion, of which the metallic minerals group was the major contributor

(43%), followed by the coal industry (29%) (Australian Bureau of Statistics, 1998).

Figure 1.2 shows each mineral products group share of the Australian export dollar for

1996-1997. The statistics from the previous years show a similar share for the metallic

minerals group. The corresponding statistics for the last five years have been

summarised in Table 1.1. This shows an average of approximately $Aus 12 billion per

annum, and 42.9% of the total mineral production.

Chapter One: Introduction 1-2

90000

80000

70000

o 60000

X 50000

2> 40000

< 30000

20000j

10000 J

Years

H Total export

D Mining industry

Figure 1.1: The Mining Industry Share of the Australian Export Dollar 1986-1998

(Source: Australian Bureau of Statistics, 1998a)

Australian mineral products by value

($AUS billion) during the year 1996-1997

8.8 13.4

9.1

H Mettalic Minerals

HI Coal

Oil & Gas

Figure 1.2: The Significance of Metallic Minerals in the Australian Mineral Products

(Source: Australian Bureau of Statistics, 1998b)


Table 1.1: Value of Australian Minerals Produced (1992-1997)*

Year

1992-93

1993-94

1994-95

1995 - 96

1996-97

Average

Total value ($m)

26,721

25,702

26,738

28,779

31,358

27,860

Metallic minerals ($m)

10,920

10,861

11,715

12,793

13,422

11,942

Metallic minerals (%)

40.9

42.3

43.8

44.5

42.8

42.9

The efficient management of producing mineral products significantly influences the

amount of Australia's income derived from exports. Hence, even minor improvements

in the production system, or reduction in the cost per tonne of the minerals, would result

in considerable profit to the industry. For example, if the enhancement of the production

management results in an increase of only one percent of the production value, the

increase in earnings will be a marginal value of $Aus 120 x 10 annually.

1.2 STATEMENT OF THE PROBLEM

Mining is a process that involves a number of stages including: exploration; ore-body

modelling; mine valuation and evaluation; mining method selection; ore extraction and

transportation; ore treatment; and finally marketing. Optimisation at each stage is

important to ensure efficient utilisation of natural resources and reduce production costs.

In this regard, many mine designers are concerned with the optimisation of the mine

geometry as it is one way of improving the efficiency of the overall mine production.

* Source: Australian Bureau of Statistics, 1998b

Chapter One: Introduction U4_

Due to the geo-technical and mining constraints, the extraction of a block of a high

grade ore may entail the extraction of some blocks of waste as well. In other words,

extra costs are incurred. For example, the extraction of an ore block in open pit mining,

requires that all materials above the block, within the pit limit, be mined first. The shape

of this pit is like an inverted cone, with the block as its base. In underground mining, the

minimum sizes of the working space may require the mining of a waste block,

alternatively, a block of ore may be left non-extracted because of the additional cost of

extracting the waste blocks surrounding the ore. However, a selected combination of

blocks may be found, as an optimum, which satisfies the exclusion of waste blocks

from- and inclusion of ore blocks to- the mine layout. The selection of a block is based

on the net economic value of the block and its neighbouring blocks. Optimisation of the

mine geometry aims to maximise the total economic value of the mine by determining

blocks for the final mine layout, subject to a number of mining constraints and

economic parameters. This means that not only are more values of ore produced, but the

cost per unit of production is decreased.

1.2.1 Status of Pit Limit Optimisation

A variety of algorithms have been proposed during the past 35 years to optimise

ultimate pit limits. The early algorithms were soon followed by some modifications to

allow those algorithms to be applied, with less restriction, and in a broader range of

mining options. The major algorithms for the optimisation of pit limits include heuristic

and mathematical approaches. Heuristic approaches include:

Moving Cone technique (Pana, 1965; and Carlson et al, 1966) and

Korobov's Algorithm (Korobov, 1974; and D o w d and Onur, 1993)

Mathematical approaches include:

Dynamic Programming technique (Lerchs and Grossmann, 1965; Johnson

and Sharp, 1971; Wright, 1987; and Koenigsberg, 1982),

Dynamic Cone method (Wilke, Mueller and Wright, 1984; and Yamatomi et

Chapter One: Introduction 1.5

al, 1995),

Graph Theory approach (Lerchs-Grossmann, 1965; Lipkewich and Borgman,

1969; Chen, 1977; and Zhao and Kim, 1992) and

Network Flow approach (Johnson and Barnes, 1988; Yegulalp and Arias,

1992; and Jiang, 1995).

In addition, artificial intelligence approaches including:

Genetic Algorithms, (GAs), (Denby and Schofield, 1994) and

Artificial Neural Networks (ANN) approach (Achireko and Frimpong, 1996)

have been suggested recently. Other miscellaneous approaches include the:

Parameterisation techniques (Lerchs and Grossmann, 1965; Whittle, 1988;

Francois-Bongarcon and Guibal, 1982; Coleou, 1986; Coleou, 1989; and

Wang and Sevim, 1992),

Bounding techniques (Barnes and Johnson, 1982; and Caccetta, Giannini and

Carras, 1986) and

Transportation algorithm (Huttagosol and Cameron, 1992)

Most of these algorithms fail to result in true optimum pit limits and approximate an

"optimum" value. However, it has been mathematically proven that the 3D Graph

Theory algorithm, and the network flow algorithm, guarantee true optimisation of

ultimate pit boundaries (Whittle and Rozman, 1991).

Mining engineers are well assisted, in pit limit optimisation, with a variety of

commercial computer packages, including those that implement the well known moving

cone and Lerchs-Grossmann algorithms. The commercial packages for the optimisation

of open pit limits include:

Minex-3D of Engineering Computer Services (ECS),

Lynx Geosystems,

Minesoft,

Mintec,


The Miner System,

MineMap,

Whittle Three-D, Whittle Four-D and

L-TOPS.

In essence, the study of pit geometry optimisation has now reached saturation level.

Many algorithms, using a vast range of techniques, have been developed. The true

optimum solution is guaranteed and several computer packages are available to the

industry.

1.2.2 Status of Stope Geometry Optimisation

Unlike open pit mining, there is a limited amount of available material for the

optimisation of the stope layout. A summary of 62 publications, which contribute to

mine geometry optimisation, is presented in Table 1.2. The major sources for the survey

include: various proceedings of the international symposiums on the Application of

Computers in the Minerals Industry ( A P C O M ) ; transactions of the Institution of Mining

and Metallurgy; and proceedings of the conferences on Mine Planning and Equipment

Selection. The results of the 62 papers clearly show there is a dearth of techniques for

optimising underground mine geometry.

Table 1.3 summarises the 62 papers on mine geometry optimisation, which are

categorised by subject. The subject areas include: general mine layout optimisation; the

introduction of algorithms; development of software for an algorithm; case study

applications of the algorithms; and reviews of the existing algorithms.


Table 1.2: Publications which Contribute to Mine Geometry Optimisation

(open pit versus underground)

#

1

2

3

4

5

6

7

Total

Period (years)

-1970

1971-75

1976-80

1981-85

1986-90

1991-95

1996-

Open pit limits optimisation

(number of papers)

4

1

3

6

10

21

7

52

Stope geometry optimisation

(number of papers)

-

-

1

3

2

4

-

10

Total

4

1

4

9

12

25

7

62

Apart from the quantity of research, the quality of the works carried out and the tools

available for optimisation of underground metalliferous mines, are as expected, much

lower than that for their open pit counterpart. Criteria, in this regard, include the status

of the developed algorithms for both methods. The proposed algorithms for open pit

limit optimisation are numerous and versatile, approximately two and a half times more

than that of underground methods. There are approximately five different algorithms

available for the optimisation of stope boundaries, including:

Dynamic Programming Algorithm (Riddle, 1977),

Downstream Geostatistical Approach (Deraisme et al., 1984),

Octree Division Approach (Cheimanoff, Deliac and Mallet, 1989),

Floating Stope Algorithm (Alford, 1995) and

Branch and Bound Technique (Ovanic and Young, 1995).

Chapter One: Introduction U8

Table 1.3: Reports on Mine Geometry Optimisation (by subject)

#

1

2

3

4

5

6

Total

Subject

General

Introduction of an algorithm

Introduction of software

Application of an algorithm

Review of algorithms

Others

Number of algorithms

developed

Number of papers

Open pit

6

26

5

8

4

3

52

12

Underground

3

5

2*

-

-

2

10

5

Total

9

31

5

8

4

5

62

15

There are various reasons for the lack of research in underground mine optimisation.

These include generality, complexity and acceptability (Ovanic and Young, 1995).

Generality: Unlike open pit mining, there are a variety of mining methods

for underground mines. Each mining method has its own conditions and

limitations. Therefore, it is difficult to develop a unique algorithm to

optimise stope boundaries for the various mining methods.

Complexity: Geological, geotechnical and economic data tend to be quite

complex in underground mines. There are no simple mathematical

formulations for many of the design problems in underground mines.

* The papers introduce both an algorithm and its software implementation.


Acceptability: Although C A D systems have automated the steps,

underground mine design practitioners are loyal to the traditional techniques

of applying 'rules of thumb' to plans and sections.

The lack of comprehensive computer-based planning tools for underground mine

planning, until the last few years, has influenced the situation in underground

metalliferous mine planning (Alford, 1995). Software developers used their skills to

tackle underground mine design, planning and production problems, after they had

successfully applied computer techniques to open pit mining applications (Foley, 1992).

1.3 SCOPE OF THE THESIS

This thesis aims to develop and implement a 3D heuristic algorithm, termed the

"Maximum Value Neighbourhood" (MVN), to optimise layouts of underground

metalliferous mines. The MVN algorithm, developed in this thesis, is implemented on a

fixed economic block model and provides a 3D analysis of the stope geometry

optimisation problem. A schematic view of the whole system, on which the thesis is

based, is illustrated in Figure 1.3. The whole study is performed in three stages, that is:

development of the "neighbourhood" (NB) concept,

development of the "Maximum Value Neighbourhood" (MVN) algorithm, on

the basis of the neighbourhood concept, and

development of a Fortran 90 based program, the Stope Limit Optimiser

(SLO), to implement the MVN algorithm. The SLO was developed with a

user interface software developer, Winteracter.

The neighbourhood (NB) concept is based on transforming the minimum allowable size

of the stope, into discrete blocks in the three dimensions. Information concerning a

block model, together with that of the stope geometry constraints, were used to develop

the concept of the neighbourhood. Discrete blocks were defined in the block model. The

geometric constraints defined the minimum stope size in the three directions. The

neighbourhood concept, for the individual blocks, is then the basis for the whole


system. That is, the MVN algorithm was developed on the basis of the

neighbourhood concept. Various possible neighbourhoods for a block are determined

and evaluated to locate the Maximum Value Neighbourhood of the block. The MVN

algorithm locates the best neighbourhood of each block, in a block model, and combines

them to form the final optimum stope.

Stope Limits Optimiser

(SLO)

MVN Algorithm

Neighbourhood Concept

Figure 1.3: A Schematic View of the Proposed Study

Finally, an interactive system, the Stope Limit Optimiser (SLO), was developed to

implement the MVN algorithm. The Stope Limit Optimiser is an application program,

developed by integrating Fortran 90 codes and Winteracter subroutines to provide a

user friendly system. The features associated with this version of the high level

language, Fortran 90, have considerably improved the capabilities of the language. The

Chapter One: Introduction

major improvements in Fortran 90 include the definition of allocatable (dynamic)

arrays, pointers and targets, enabling the definition of derived types, case structure and

use of modules. A user interface environment was developed by employing the features

of Winteracter version 1.10. Winteracter and Interacter are two products of Interactive

Software Services Ltd. (ISS), which supply software development tools for visual

interface by the user. Interacter provides a DOS environment for the development of

software while Winteracter is based on Windows environment. Winteracter provides

menus and dialog boxes, through which the user can select options, input required data

and retrieve outputs.

1.4 OUTLINE OF THE THESIS

The thesis has been designed in nine chapters. The first (current) chapter is to introduce

the research. Chapter T w o discusses the ultimate mine optimisation methods in general

and reviews the major existing optimisation algorithms for both open pit limits and

stope boundaries. It also reviews briefly the process of constructing an economic block

model of an ore-body. Chapters Three and Four correspond to the development of the

MVN algorithm. Chapter Three discusses the basics of the neighbourhood concept and

the MVN algorithm in one dimension, while applying the algorithm on simple examples

manually. Chapter Four reviews generalisation of the MVN algorithm for 2 D and 3D

neighbourhoods. Application of the 2 D algorithm on a simple example is also provided.

In addition, implementation of the algorithm on a small sized 3 D example is introduced

using Visual Basic for Applications (VBA) macros of Excel, to test the methodologies.

Chapters Five, Six and Seven cover the development of the Stope Limit Optimiser

(SLO). Chapter Five presents an overview of the SLO system and explains how the input

of the optimiser is provided, while Chapter Six discusses the implementation of the

optimisation algorithm and Chapter Seven discusses the outputs of the SLO. Chapter

Eight discusses h o w the SLO was validated and Chapter Nine gives the conclusions and

suggestions for future work.

CHAPTER TWO

ULTIMATE MINE DESIGN METHODS

2.1 INTRODUCTION

The stages of mining begin with exploration and end with the production of a

commercial product. The intermediate stages of mining include: ore-body modelling;

mine valuation; mine evaluation; mining method selection; ore extraction; ore

transportation; ore treatment; and marketing. In general, attempts are made to balance

each stage in the most optimum manner. A number of algorithms have been developed

in the past for the optimal determination of the ultimate mine geometry for both the

open pit and underground mining methods. These optimisation algorithms are mostly

based on computerised methods and are performed on a geological model. When using

these algorithms, care should be taken that it is the model that is optimised and not the

real ore-body (Kim, 1978). That is, the accuracy of the optimisation depends on a

reliable representation of the ore-body. This chapter reviews the existing algorithms for

the optimisation of the open pit limits and underground stope boundaries. A review of

computerised ore-body modelling, required for optimisation algorithms, is also

presented in this chapter.

Chapter Two: Ultimate Mine Design Methods . 7/2

2.2 MINE GEOMETRY OPTIMISATION

The definition of the mine layout is one of the most important stages of mine planning

for both surface and underground mines. Outlining the mineable ore assists in the

determination of the amount of the reserve, as well as the mine life and production

scheduling. There is a mutual relationship between mine layout definition and other

mine planning stages, such as: equipment selection; haulage routes; and bench height.

This means mine design is an interactive process with a paradox between mine layout

definition and production scheduling. According to Whittle and Rozman (1991), prior

to working out the mining schedule, the optimal layout of the mineable ore has to be

defined. The optimal layout cannot be determined until the values of the blocks are

known. However, the block values depend on factors, such as the commodity price and

mining costs (blasting and transportation), which in turn depend on when the blocks are

to be mined. Despite this fact, the mine layout optimisation and the production

scheduling are usually addressed separately (Gill, Robey and Caelli, 1996), which leads

to the use of certain assumptions and approximations. In this thesis, only, mine layout

optimisation is covered.

2.2.1 Optimisation Criteria

Mine geometry optimisation requires using the formal procedures of Operations

Research (OR). These include: the formulation of the problem; the definition of the

objective function; and the formulation of constraints. The following objective function

criteria have been used in mine geometry optimisation (Wright, 1990):

1. maximisation of the total mine economic value,

2. maximisation of value per tonne of the saleable product,

3. maximisation of the life of the mine, provided the value per tonne does not

fall below a certain figure and

4. maximisation of the metal content within the mine.

The most frequently used criterion is the maximisation of the total mine economic

Chapter Two: Ultimate Mine Design Methods 2-3

value. Wilke, Mueller and Wright (1984) have shown that the variation of the total

value with the size of the pit (tonnage of the saleable product) is different from that of

the unit value (Figure 2.1), for a specific mine.

Value

G = total value ($) g = specific value ($/t)

Tonnage of the saleable product

Figure 2.1: Variation of the Total Value Compared to the Unit Value of a Specific Mine

over the Mine Production (Wilke, Mueller and Wright, 1984)

The total value and the unit value curves in Figure 2.1, show the value per tonne of the

saleable product, reaches its maximum at a lower amount of production when compared

with the total economic value of the mine. That is, the optimal mine layout is greatly

influenced by the chosen optimisation criterion.

2.2.2 Problem Formulation

Taking the maximisation of the total mine economic value as the optimisation criterion,

the problem is formulated to find the mine outline which has the maximum total

economic value. In order to achieve this, an economic block model of the ore-body is

required. The problem is reduced to selecting those blocks within the economic model

that maximise the total value, while the selection of the blocks is constrained by the

geometry of the mine (pit or stope). The problem is, then, simplified to find (Gill,

Robey and Caelli, 1996):


' 0,J,k)ey

where

BEVyk: the block economic value and

y: the set of blocks making up feasible mine geometry design.

The principal method of mining defines the optimisation constraint. In open pit cases,

the mine geometry follows an inverted cone shape. The constraint is, therefore, the

ma x i m u m allowable slope angle of this cone. However, the geometry constraint, in the

optimisation of the stope boundaries follows the cubic shape of the stope. This can be

formulated in terms of the minimum stope dimensions.

The optimum mine layout, if found, is unique. Since the optimal solution is the one

with the highest dollar value, there is no single block, or combination of blocks, which

can be added to, or removed from this mine layout that can produce an increase in the

total value of the mine, within the specified layout. Assuming that there are two layouts

with the same value implies that none of the layouts is optimal, since if taken together,

the two layouts would produce an outline of higher value (Whittle, 1990).

2.2.3 Necessity of Optimisation Algorithms

The optimal mine layout is usually given in terms of a list of blocks, selected from an

economic block model. It is necessary to determine whether or not every single block

should be included in the optimal list of blocks. That is, there are two options for each

block, selected or not selected. The number of possible alternatives for a combination of

n blocks within the model is equal to 2n, one of which is the optimal. A trial and error

approach to find the unique optimal layout out of 2n alternatives, even for a very small-

sized model, can take millions of the time unit (Whittle, 1993). For example, a 2D

section of 10 x 10 blocks requires a check of 2m (approximately 1030) alternatives.

Therefore, it is necessary to develop optimisation algorithms based on practical mining

constraints. These algorithms must provide a search rule or a mathematical formulation,


which excludes most possible alternatives from the calculations. For example, a

heuristic algorithm which searches ore blocks, can considerably reduce the possible

alternatives. Considering that ore blocks may constitute 3 0 % of the whole block model,

the number of alternatives would decrease significantly from 210 0 to 230 alternatives

(that is, from 1030 to 109 alternatives).

2.3 ULTIMATE PIT OPTIMISATION ALGORITHMS

A variety of algorithms have been developed during the last decades to determine the

shape of the ultimate pit. Lerchs and Grossmann (1965) and Pana (1965), who

independently studied the same problem, can be considered the pioneers in this field.

Kim (1978) has categorised the ultimate open pit limit design methodologies into two

main groups: rigorous and heuristic. The term rigorous was used to imply the

availability of mathematical proof for the optimising technique while the term heuristic

was used to describe an algorithm which works in nearly all cases but which lacks

rigorous mathematical proof. Graph Theory, Network Flow, Linear Programming and

Dynamic Programming algorithms fall into the rigorous category while the Moving

Cone technique and the algorithm of Korobov were considered heuristic. More recently,

Gill, Robey and Caelli (1996) have categorised the algorithms into historical

approaches, including Moving Cone and Dynamic Programming and modern

techniques, which included Graph Theory and Network Flow solutions.

Thomas (1996) has presented five parameters to categorise the existing algorithms.

These parameters include:

1. "rigorous", a term applied to techniques which always find the optimum

solution if given sufficient time (like the Dynamic Programming technique),

2. "heuristic", a term applied to any technique which is not rigorous and finds

only an approximate solution, regardless of being close to the true solution or

not (like the Moving Cone technique),


3. "stochastic", a term applied to techniques whose analyses are based on

probabilistic sampling from the range of possible solutions (like the genetic

algorithms),

4. "static", a term applied to analyses which ignore the effects of time on the

valuation process (like the Lerchs-Grossmann algorithm) and

5. "dynamic", a term applied to analyses which take into account the effects of

time (like Whittle Four-D).

Based on the above classification, the Graph Theory algorithm of Lerchs and

Grossmann would be described as rigorous and static, whereas a genetic algorithm,

which included financial discounting effects would be described as dynamic, stochastic

and heuristic.

However, there is no global agreement on the classification of optimisation algorithms.

The major existing algorithms, for optimisation of ultimate pit limits may be listed as:

Moving Cone technique (simulation approach),

Korobov's Algorithm,

Dynamic Programming technique,

dynamic cone method,

Graph Theory approach (Lerchs-Grossmann algorithm),

Network Flow approach,

Linear Programming technique,

parameterisation technique,

bounding technique,

transportation algorithm,

genetic algorithms (GAs) and

artificial neural network ( A N N ) approach.

The most popular approaches used in open pit limit optimisation from the above list

include the Moving Cone technique, Dynamic Programming algorithms and the Graph

Theory algorithm. These three algorithms, together with their modified versions, are

Chapter Two: Ultimate Mine Design Methods

briefly reviewed in the following sections.

2.3.1 Moving Cone Algorithms

The Moving Cone approach and the Korobov algorithm may be considered heuristic.

The Moving Cone technique is the simplest and fastest algorithm developed for pit limit

optimisation. The algorithm proposed by Korobov is a major improvement of the

technique. The Dynamic (Moving) Cone algorithm, which is a combination of applying

both Moving Cone and Dynamic Programming techniques, may be considered a

heuristic algorithm as well.

The Moving Cone algorithm, as described by Carlson et al. (1966), is based on

constructing an inverted cone above any ore block (the block with positive economic

value). The cone includes all blocks that should be removed preceding the mining the

ore block. The angle of this cone is controlled by the ultimate slope angle, which

satisfies the stability of the pit and may vary in different directions. To be more precise,

it is usually not a true cone, but a pyramidal approximation of an irregular cone (Gill,

Robey and Caelli, 1996).

The Moving Cone algorithm uses a simulation approach to determine the optimal pit.

The basic element to simulate the mining of the pit is the minimum removal cone (the

inverted cone based on the block under consideration). Steps taken in the algorithm as

stated by Wright (1990) are as follows:

1. Start from the surface and search for ore blocks (blocks with positive

economic values).

2. Construct the minimum removal cones on such ore blocks.

3. If the cone value (the sum of block economic values, BEV, of all blocks

contained in the given cone, including the ore block in question) is positive,

consider the cone removed (that is, mining is simulated).

4. Continue the search until all the ore blocks in the block model have been


examined.

5. The ultimate pit is formed by the shape left after removal of all positive

valued cones.

Figure 2.2 illustrates the procedure used by the Moving Cone technique to the ultimate

pit optimisation, where the m a x i m u m slope angle of the pit is 45.

The Moving Cone algorithm benefits from its simplicity in both concept and computer

implementation. Its main problem is how to deal with a group of blocks, which fall

inside more than one cone, that is, the problem of overlapping cones. In the example

shown in Figure 2.2c, it can be easily seen that after the algorithm is completed, the pit

is not maximised. That is, the inverted cone based on the block, B27, which is valued at

+1, can be added to the pit and make a pit valued at +3. Figure 2.3 illustrates the failure

of the Moving Cone to recognise the maximum valued pit. This also shows that if the

block, B44, is examined before the block, B27, the technique leads to a different pit value.

In general, as stated by Lemieux (1979), the order of the search for examining ore

blocks plays a very significant role in the Moving Cone algorithm.

In addition, two separate cones may be of negative value, yet if taken together, a positive

pit is obtained. In these cases, the Moving Cone technique fails to recognise the positive

valued pit. As illustrated in Figure 2.4, both cones based on ore blocks, B33 and B35, are

of negative value, and the Moving Cone technique would conclude that there is no pit at

all. However, by combining the two cones, a pit of value +1 can be obtained (Figure

2.4c).

Chapter T w o : Ultimate Mine Design Methods 2-9

> J

(a)

-1

-1

-2

-2

-1

2

-2

-2

-1

4

-2

-2

-1

0

-2

8

-1

1

-2

-2

-1

0

-2

-2

-1

2

-2

-2

-1

-1

-2

-2

A section example

o>>

1

2

3

-1

C = -l

-1

-1

-2

-2

-1

2

-2

-2

-1

4

-2

-2

-1

0

-2

8

-1

1

-2

-2

-1

0

-2

-2

-1

2

-2

-2

-1

-1

-2

-2

c = -i

-1 -1 -1 -1

Pit value = 1

C = -\

(c)

-1

-1

-2

-2

I

2

-2

-2

C M - '

I "2

-2

^

0

-2

8

-1

1

-2

-2

-1

0

-2

-2

-1

2

-2

-2

-1

-1

-2

-2

Pit value = 2

Note: C is the cone economic value based on the block, By.

Figure 2.2: The Moving Cone Technique

2-

-/ [ c27=i

-1

-2

-2

-2

-2 -2 -2

-2

-2

2

-2

-2

-1

-1

-2

-2

Pit value = 3

Figure 2.3: Failure of the Moving Cone to Recognise the Maximum Value Pit

> J

1

2

3

1

-1

-2

-3

2

-1

-1

-3

PF=-2

3 4

-1

-1

6

-1

-1

-2

5

-1

-1

7

6

-1

-1

-3

7

-1

-2

-3

1

2

3

1

-1

-2

-3

2

-1

-1

-3

3

-1

-1

6

4

-1

-1

-2

5

-1

-1

7

6

-1

-1

-3

7

-1

-2

-3

(a) (b)

1

2

3

1

-1

-2

-3

2

-1

-1

-3

PV=+]

3 4

-1

-1

6

-1

-1

-2

5

-1

-1

7

6

-1

-1

-3

7

-1

-2

-3

Note: PV= Pit economic Value (c)

Figure 2.4: Failure of the Moving Cone Algorithm to Recognise a Positive Value Pit

(Wright, 1990)

Modifications, including Korobov (1974), have been suggested to overcome the


problem of overlapping cones. The Korobov algorithm was based on evaluating the

positive net value blocks in levels, level by level, from the top to the bottom. Within

each cone, the algorithm allocated the positive valued blocks against the negative valued

blocks until all negative valued blocks can be combined with the positive ones. The

algorithm required several passes to repeat the process. However, the earliest version of

the Korobov algorithm, as described by David, D o w d and Korobov (1974), failed to

overcome the problem of overlapping cones in all conditions. A major enhancement to

the Korobov algorithm was suggested by D o w d and Onur (1992), in which they

proposed the joint testing of cones before it was decided whether to remove a cone or

not. The latest improvement in the Moving Cone technique, called Moving Cone II, may

be found in Wright (1999).

2.3.2 Dynamic Programming Algorithms

Application of a Dynamic Programming (DP) technique to the optimisation of the

ultimate pit limits was first introduced by Lerchs and Grossmann (1965). The first D P

algorithm was simple and rigorous but suitable for only 2 D cross sectional problems.

Smoothing routines were then developed to overcome this shortcoming of the 2D D P

algorithm. Other modifications contain the development of algorithms to directly apply

3 D D P technique.

The basis of the 2D DP approach is illustrated in Figure 2.5. The block model is first

divided into several slices or parallel vertical sections, which are arranged in /rows and

./columns. The economic value of blocks (BEV) are represented by the term mip shown

inside each cell (Figure 2.5a). To determine the optimum contour of each section, three

straight lines must be defined. That is, the bottom of the pit and two walls, at a slope

angle of a, which is the maximum angle allowed to meet the slope stability requirement.

For simplicity, the earliest algorithms assumed that a is constant over the whole pit.

Width and height of blocks define the pit slope, a, that is:


Block Height

Block Width = tan (a)

In other words, the maximum pit slope constraint is satisfied by moving one block left

(right) and one block up (down). In order to mine a block (in Figure 2.5a), all blocks

above it must be removed prior to mining the block. The summation of the values of the

blocks of the same column, from the surface to and including the block under

consideration, defines the cumulative economic value of that block. This is termed My,

that is:

i Mij = m / for; =1,2,...,/ (2.2)

n=\

where

mnf. the economic value of the block located in row n and columny

J: the number of columns in the section and

Mij: the cumulative economic value of a column of blocks from the surface to and

including the block, By.

The value My is computed and assigned for each block (Figure 2.5b). The mining of the

block, By, requires removing all blocks which fall within the minimum cone, that

satisfies the slope constraint. That is, the block must be extracted jointly with one of its

neighbouring blocks from the previous column, either from the previous row (/.;, j.i),

same row (Bu j.i), or subsequent row (Bi+], j.i). The optimum contour requires that the

one which provides the maximum pit value, be selected to be considered jointly with the

block, By. Hence, another term, Py, is used to represent the pit value when the pit is a

cone based on the block, By. The pit value corresponding to each block is computed

through a recursive formula (Equation 2.3).

"'-1,7-1

P..=My+max\PiiH (2.3)

Pi+IH


where Py is the maximum possible contribution of columns 1 to j to any feasible pit

that contains the block, By, on its contour.

+ J

(a)

-1

-1

-2

-2

-1

2

-2

-2

-1

4

-2

-2

-1

0

-2

8

-1

1

-2

-2

-1

0

-2

-2

-1

2

-2

-2

-1

-1

-2

-2 m block net value

(b)

0

-1

-2

-4

-6

0

-1

1

-1

-3

0

-1

3

1

-1

0

-1

-1

-3

5

0

-1

0

-2

-4

0

-1

-1

-3

-5

0

-1

1

-1

-3

0

-1

-2

-4

-6 M block cumulative net value in the column

(c) 2

3

4

- o<

N

*

-

-

- 0*4

\

-

- o<

V,

-34

M fr

- 0

24 / * - 2

\

^4

2-* /

\

4 - 2

- 2A

1 /

33-/

- 1

-

- 2A

2 /

- 4 *

-

-

- 2

3*

-

-

3 /

Pij

Figure 2.5: A Dynamic Programming Technique Applied to Pit Limit Optimisation


The formula can be modified to a general form as expressed by Equation 2.4:

Py =MiJ+max{Pi+rH} (2.4)

where r is the range of rows of blocks, of the previous column, that should be included

in the neighbouring blocks, to satisfy the maximum slope constraint.

Determining the optimum neighbour does not necessarily cause its inclusion in the

optimum pit, but rather implies that if the examined block is a boundary block, it has to

be mined jointly with its optimum neighbour. The algorithm starts with adding a new

row on the top of the section (that is, i = 0), as shown in Figure. 2.5b. This artificial row

contains air blocks with zero pit values (that is, PQ/ = 0), and helps in maximising the pit

value of neighbouring blocks when processing blocks in the first row. The pit value,

corresponding to each block, Py, is computed starting from the block, Boo, as in Figure

2.5c. Blocks are examined row by row within each column then column by column

within the section. After determining the optimum neighbour for each block, an arrow is

drawn from the examined block pointing to its optimum neighbour (Figure 2.5c). After

all blocks are examined, the optimum pit is obtained by maximising the pit value

computed for blocks in the artificial row (that is, i = 0), as expressed in Equation 2.5.

?max=maxP0y for j = 1, 2, ..., J (2.5)

If the maximum value of P in the artificial row is positive, then the optimum contour is

obtained by tracing the arrows from, and to the left of, the block containing this

maximum. In cases, where the maximum value of? in the first row is negative, there is

no pit containing a positive value. Figure 2.5c illustrates the determination of the

optimum contour for the section by computing the P value and the optimum neighbour

of each block.

The 2D DP algorithm provides the true optimum contour for each section. However, it

is impractical in the real 3D pit, because when all the optimised contours of the vertical

Chapter Two: Ultimate Mine Design Mthu ' 2-15

sections are set to form the final pit, it appears that they do not fit together. In other

words, while cross-sections have been optimised subject to maximum slope angle, a,

longitudinal sections do not necessarily satisfy this constraint.

Some modifications have been proposed for smoothing the pit contour so that the slope

limit requirement be met. The algorithm of Johnson and Sharp (1971) was based on

double optimisation, that is, once for cross sections and another time for right angles to

the original sections. Their algorithm does not solve the problem completely and suits

the elongated deposits. Another smoothing routine is the Dynamic Path approach

suggested by Wright (1987). The Dynamic Path approach exploits two facts from the

Moving Cone and 2 D D P techniques. The first is that the 2 D D P technique guarantees

the true optimum contour for vertical sections and the second is that the 3D Moving

Cone approach automatically supports the boundary smoothing. The Dynamic Path

algorithm can solve the smoothing problem completely. It applies the same 2D recursive

formula (Equation 2.4) in a forward pass while changing the direction of analysis when

moving from one section to another one. The union of the minimum removal cones

along the Dynamic Path defines the ultimate pit. Despite the fact that smoothing

routines provide speed and small computer memory requirements, they lack the ability

to guarantee the true optimal solution (Wright, 1990).

Other attempts have been made to directly apply the DP technique to optimise the 3D pit

limits. These include the algorithm of Koenigsberg (1982) and the Dynamic Cone

algorithm, which have been developed mostly by modifying the 3D neighbouring

blocks. In the 2 D D P algorithm a given block can be accompanied with only one

neighbouring column in the backward direction, that is, the column which was

previously examined. However, in a 3 D problem, there are four neighbouring columns

in the backward direction. Koenigsberg (1982) addressed these columns as:

Si in column (j-1, k): side of i

BSi in column (j-1, k-1): back of side of i

SBSi in column (j, k-1): side of back of side of i

SSBSi in column (j+1, k-1): side of side of back of side of i


The recursive formula was then modified accordingly, to cover maximisation among

all the neighbouring blocks. Wilke, Mueller and Wright (1984) addressed the

degeneration problem of the Koenigsberg's algorithm and developed a new recursive

formula, which was based on the use of 3D increments, in the form of minimum

removal cones over each block. Such an approach uses the Moving Cone technique in

the D P application and has been labelled as the Dynamic Cone algorithm. A recent

modification of the dynamic cone algorithm, the selective extraction dynamic cone

algorithm, was introduced by Yamatomi et al. (1995). The improvement is based on

examining whether or not: any negatively valued blocks are included in a unit cone; it is

possible to leave the negatively valued blocks unmined in the unit cone, without

disturbing the geometric condition imposed on the slope angle of the cone.

2.3.3 Graph Theory Algorithms

The application of Graph Theory (GT) to the optimisation of the ultimate pit design was

first introduced by Lerchs and Grossmann (1965) as a result of the impracticality of their

2 D D P algorithm in three dimensions. The technique did not receive much attention for

many years, mostly due to some computing time problems. With improvement in

capabilities in computers, the application of the Graph Theory technique was again

considered. The early algorithm was improved and commercial computer packages were

developed (Lipkewich and Borgman, 1969; Chen, 1977; Zhao and Kim, 1992; Alford

and Whittle, 1986). The 3D G T algorithm is widely known as the Lerchs-Grossmann

(LG) algorithm. Alternative approaches to the L G Graph Theory algorithm have also

been proposed. These include the mathematical programming solution suggested by

Picard (1976) and the solution derived from a Dual Simplex viewpoint described by

Underwood and Tolwinski (1996).

The LG algorithm is based on transforming the block model into a weighted directed

graph and finding the maximum closure of the graph (Lerchs and Grossmann, 1965). A

directed graph, G = (V, A), is defined by a set of elements, V = (v,, v2> .... vnJ, which are

called vertices of G, as well as a set of ordered pairs of elements, A = (a, a2, -,an}


where ak = (vb Vj), which are called arcs of G. In a physical interpretation, an arc is

an arrow, which connects to vertices, pointing from the first vertex (predecessor) to the

last vertex (successor). A closure of G is defined as a set of vertices Y c: V such that if

V, e Y and (vh vj) e A, then v,- e K In other words, if v, is in the closure, then all

successors are in the closure as well. W h e n a real number, mh which can be negative,

zero, or positive, is associated to each vertex, vt, of the graph, G, which is called the

mass of v then a m a x i m u m closure of G is defined as a closure of maximum mass. As

stated by Picard (1976), the main application of the maximal closure of a graph is found

in the ultimate pit optimisation.

The application of the GT to the optimum pit, as formulated by Lerchs and Grossmann

(1965), follows. Let the entire pit be divided into a set of blocks, Bh regardless of being

regular or irregular blocks. Each block is associated with a mass, m representing the net

economic value of the block. The block, Bh is adjacent to Bj, if it has at least one point

in c o m m o n with Bj. (Figure 2.6) The block, Bh is also dependent on Bj, if removing Bt is

not permitted unless Bj is mined already. Transforming blocks, B, to vertices, V, an arc

(vh vj) is drawn if Bt is adjacent to and dependent on Bj. Hence, a 3D directed graph G =

(V, A) with a set of vertices, V, and a set of arcs, A, is obtained. Any feasible contour of

the pit can then be represented by a closure of G. Considering the net value of each

block as the mass, mh associated with its corresponding vertex, the problem is

formulated in finding the maximal closure of G. The L G algorithm, to find the

m a x i m u m closure of a graph is based on a tree structure, starting with the construction

of a tree, 1, in G. The Branches of trees, which are defined by arcs, represent the

relationships between the blocks, that is, whether or not the ore blocks can support the

attached waste blocks. Based on graph rules, the tree, f is transformed into successive

trees, T},f,..., T, until there is no possible transformation (Lerchs and Grossmann,

1965). The vertices of a set of well-identified branches of the final tree will then define

the m a x i m u m closure of G and represent the pit with the maximum profit.

The LG algorithm is able to ensure the true optimality of pit design. It can also handle

the pit optimisation of irregular block models. The implementation of the algorithm


requires too much computing time and memory. The connection information of a

medium-sized model (150,000 blocks) requires over a gigabyte of storage space or

virtual memory (Gill, Robey and Caelli, 1996).

> j

-1

-1

-2

-2

-1

2

-2

-2

-1

4

-2

-2

-1

0

-2

8

-1

1

-2

-2

-1

0

-2

-2

-1

2

-2

-2

-1

-1

-2

-2

(a) A section of a block model

A closure

rn fm m m m rn rn n^ rn m -00 ( -1 -1 -1 -1 -1 -1 -1 ) -1 -00

m 1 \ 9 4 0 1 0 /' 9 -1 m

m -9 -2 N\ -2 -2 -2 / -2 -2 -2 -oo

f-2] -2 -2 \ 8 / -2 -2 -2 -2

(b) A graph theoretical representation of the block model

Figure 2.6: Graph Theory Applied to Pit Limit Optimisation

Lipkewich and Borgman (1969) suggested the evaluation of the L G model using sub-

graphs in separate levels, starting from the top and working to the bottom of the model.

The L G algorithm was implemented for each level and the optimised sub-pit was

removed before proceeding to the next level. Chen (1977) addressed the difficulty of the

L G algorithm in calculating an optimum pit with variable wall slopes. H e proposed a


search pattern for handling this problem.

Zhao and Kim (1992) developed a modified LG algorithm, which was simpler than the

well-known L G algorithm to implement. The algorithm of Zhao and K im was aimed at

considerably reducing the number of generated arcs. In their algorithm, arcs were only

generated between either ore to waste vertices, or waste to ore vertices. That is, no arcs

were generated between two ore vertices, nor two waste vertices. Their algorithm was

claimed to be faster than the L G algorithm, although the claim has been questioned

(Gill, Robey and Caelli, 1996).

Alford and Whittle (1986) discussed the computer implementation of the LG algorithm

in one package for designing open pit mines. Their package was later improved and is

commercially marketed as Whittle Three-D. The latest version, Whittle Four X, is a

highly sophisticated package, which allows for economic analysis and simulation of

long-term mining projects.

2.3.4 Network Flow Analysis

The application of the Network Flow (NF) technique to the pit limit optimisation was

first proposed by Johnson (1968). A mathematical support for the N F analysis can be

found in Picard (1976). Johnson and Barnes (1988) applied the earliest maximal flow

algorithms to design pits. The implementation of the latest algorithms has been reported

by Yegulalp and Arias (1992) and Jiang (1995). Giannini et al. (1991) developed the pit

design software, PITOPTIM, incorporating a high speed Network Flow technique.

A network consists of nodes, arcs and values associated with arcs, which are called

capacities. Figure 2.7 illustrates a typical network model. In constructing such a

network model the following steps are undertaken (Jiang, 1995).

1. Represent each block, within the block model, by a node in the network

model.

Chapter T w o : Ultimate Mine Design Methods 2-20

2. Add two artificial nodes to the network model, that is, a source node and

a sink (terminal) node.

3. Generate arcs from the source node to all nodes corresponding to the positive

valued blocks. Each arc is assigned a maximum flow capacity that is equal to

the net economic value of the corresponding block (BEV).

4. Generate arcs from all nodes corresponding to the negative, or zero valued

blocks, to the sink node while assigning the corresponding BEVs as

capacities of those arcs.

5. Generate arcs from all nodes with positive valued blocks to nodes which are

negative or zero valued blocks. Non-positive valued blocks are to be mined

if the positive valued block is mined.

-> J

1

2

3

1

-3

-4

j

2

-1

+3

-4

3

4

0

+5

4

+1

+2

-2

j

5

-2

1 1 1

-2 ! j

(a) A section example

Positive valued blocks

Non-positive valued blocks

Source

4 /

^ " ^ 3

5 \

w w

4

1

3

2

5

tr Infinite capacity

(b) A network representation of the block model

Figure 2.7: The Network Flow Technique Applied to Pit Limit Optimisation


The objective of network analysis is to maximise the amount of flow from the

source node to the sink node, knowing that all arcs are directed to the sink node. That

is, the flow through the network is permitted only in the direction of the sink node. The

flow through the network from the source to the sink represents the transfer of the

economic value away from the ore blocks, as these are required to cover the cost of the

removal of the related waste blocks (Dincer and Golosinski, 1993). W h e n the maximum

flow solution is obtained, arcs may or may not reach their maximum capacity

(saturation). Saturated arcs show that the corresponding positive valued blocks are

exhausted and cannot contribute to the final pit value. The unsaturated arcs from the

source (those with excess capacity) can identify the optimum pit limits. The positive

valued blocks corresponding to these arcs, together with all the negative or zero valued

blocks on which these blocks are dependent, can be mined at a profit.

Several algorithms have been developed to maximise flow through the network. The

earliest one was developed by Ford and Fulkerson (1956) and is called a labelling

algorithm. Yegulalp and Arias (1992) have listed 18 algorithms for the maximisation of

the flow through the network.

The Network Flow analysis benefits from a relatively simpler concept and algorithm

(compared to Lerchs-Grossmann (LG) algorithm) and guarantee the true optimal

solution to the pit limits optimisation. For large-sized block models, the Network Flow

analysis is not suitable, although it may provide faster solutions for ore-bodies with a

small number of blocks (Yegulalp and Arias, 1992).

2.3.5 Other Approaches

In addition to the above algorithms, the application of a wide range of techniques to pit

geometry has been proposed. Some of these approaches have been devised for the joint

consideration of ultimate pit limits determination and extraction scheduling. Some

others aim to directly solve the ultimate pit limits problem. There are approaches which


aim to reduce the computing time associated with the existing algorithms.

2.3.5.1 Linear Programming Approach

The application of the Linear Programming (LP) techniques to pit optimisation was

described by Meyer (1969). However, the most significant disadvantage of the approach

was its excessive need for computer memory and computing time. The ultimate pit limit

problem was formulated as a large scale transportation problem with an LP solution by

Huttagosol and Cameron (1992). The adapted simplex algorithm of the LP is used to

solve the dual system. The required computing time and memory are the main

constraints of the method, especially for a large scale problem.

2.3.5.2 Parameterisation Techniques

Due to variation of economic parameters over time, the optimum pit limits would vary

as well. The technique is, therefore, based on generating the nested pits, each of which

is obtained by running the optimisation algorithm while changing the value of each

block. In the latest improvement, Whittle (1988) defined an economic parameter as the

ratio of metal price to

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